One of the central goals in neuroscience is to "crack'' the neural code, i.e. to fully characterize how the brain understands the state of the world, and to understand the specifics of neural spiking activity which convey information. One common approach is to model spike trains as realizations of a point process. As a longer term project, I am interested in creating a large-scale point process framework for the simultaneous analysis of neural spike trains recorded from a large ensemble of neurons. This is a highly collaborative project between computer science, statistics, and experimental neuroscience. Working alongside my prospective students and colleagues in the Statistics/Mathematics department as well as those in the fields of Computer Science, Biology and Neuroscience will offer countless opportunities to understand the multiple facets of neuroinformatics, advancing the field from multidisciplinary viewpoints.
I am currently developing new methodologies for the analysis and modelling of neural spike trains within a point-process framework, which allows for two types of almost equivalent analyses:
Conditional intensity function, which is defined to be
where N(t) is the number of spikes up to, and including, time t, and Ht shows the spike times up to time t. Formulating the intensity functions as a function of the spiking history and some covariates through classical GLM models is a common approach.
Modelling the Inter-spike Interval (ISI) distribution. One approach in this category is renewal process, but more generally, the i.i.d. assumption of the ISIs within renewal processes is loosened due to non-stationarity of the neural spiking activity. Other alternatives include inhomogeneous Inverse Gaussian process, and inhomogeneous Gamma process to model ISIs.
Multiscale Analysis of Neural Spike Trains
Several studies in the literature suggest that neural spiking activity has multiple aspects/properties which belong to different time scales, ranging from milliseconds (the dynamics of the ion channels) to days or even months (learning, aging, etc.). My co-authors and I have proposed a novel and computationally fast approach for estimating the intensity function of spiking activity within an inhomogenous Poisson process framework. In this paper, we have introduced a powerful graphical technique which can visualize biological phenomena from different time scales. Also developed are multiscale models based on a wavelet-based penalized likelihood technique. Our models allow for periodic terms in the intensity function (additive or multiplicative), which accommodate the periodic spiking activity (a.k.a. brain rhythms), and/or for the effects of periodic stimulus signals. Employing this methodology on a dataset from anatomically connected retinal ganglion cells (RGC) and lateral geniculate nucleus neurons (LGN), we have shown that retinal ganglion cells and their paired LGN neurons have relatively similar multiscale models, outperforming the popular BARS model. We have also addressed several theoretical, computational, and application-related points to guide researchers through the proper use of this new methodology. It is important to emphasize that computational challenges are often present in modelling neural spike trains, however, our multiscale models are significantly faster than commonly used alternatives in the literature.
Skellam Process with Resetting (SPR)
Although Poisson models are widely employed for the analysis of neural spike trains, the serial dependence among spike times, biological phenomena such as refractory period and bursting, and lack of multivariate extensions with flexible dependency structure, motivated me to introduce a new neural spike train model within a point process framework. Furthermore, most statistical models in the literature lack biological justification, and my primary interest is developing methodologies which are biologically justified.
Neural integration is a neurobiological law stating that a neuron fires a spike when its integrated excitatory and inhibitory electrical inputs cross the threshold of excitation (-70mv). Motivated by this law, we have proposed the Skellam Process with Resetting (SPR). Skellam process is defined as the difference between two independent Poisson Processes, representing the inhibitory and excitatory inputs to a neuron. Modifying this process for the refractoriness of neurons, SPR models the spike times as the k-th record times of a Skellam process, and resets the process to state zero after each spike in order to accommodate neural refractoriness. The diagram below shows the biological spike generation process, as well as SPR as a statistical model ``mimicking'' the neural integration process.
I have studied the theoretical properties of the Skellam process, and showed that the ISI distribution within SPR framework is a mixture of Gamma distributions. This closed form ISI distribution allows for computationally efficient statistical inference. We have also shown that depending on parameter values, ISI distribution within SPR varies from exponential to Inverse Gaussian and Lognormal. It is noteworthy that Inverse Gaussian is one of the most common models for the ISIs. Based on a study on connected retinal ganglion cells and neurons from the cortex, we have shown that our model provides a good fit to the data.
Multivariate Skellam Process with Resetting (MSPR)
Simultaneous analysis of multiple spike trains is a challenging problem, and the current multivariate distributions to model multiple neural spike trains have limited dependency structure. We have generalized the results of the univariate Skellam model to the multivariate case. The multivariate Skellam vector is formally introduced and its distributional properties are studied. One of the interesting properties of the multivariate Skellam model is that unlike the multivariate Poisson distribution, which only allows for positive correlation, it does not suffer from limited dependency structure, allowing for the analysis of inhibitory neurons where clear negative correlation is observed. Multivariate Skellam processes in both homogeneous and inhomogeneous cases have been introduced. Moreover, expanding on SPR, multivariate Skellam process with resetting (MSPR) has been developed to accommodate simultaneous analysis of multiple neural spike trains. The computational algorithm from the univariate case has also been extended to the multivariate model, which allows for parallel computations on clustered computers.